Twin Paradox- a quick(ish) question

[otg]

Let me see if I understand correctly. In the upper part of the diagram, you have "A" being the Earth, at rest. "B" travels outbound from Earth, and passes "C", who travels back to Earth again. No problem there.

Correct

What is the next diagram trying to show? Is it the same events from the perspective of "C"? If so, you have "B" passing by "A" on the right, then synchronizing with "C", and then "C" ariving at "A"... except that this time, "A" is moving towards "C". So you really need to indicate "A" moving towards "C", but more slowly than "B".

No, it is the same events from the perspective of "A" [who I named "B" in the previous pic]. In your words:
In the lower part of the diagram, you have "A" being the spaceship, at rest. "B" travels outbound from spaceship, [that is, Earth moves away from "A" in "A":s perspective] and passes "C", who travels back to spaceship again.

This is [or should be] identical to the above situation. Why is it not? Why must I say that it is the spaceship that shifts frames?

The age of "A" in this series of events is unambiguous, because "A" is there at the start, and at the finish. But what age of "C" can you compare? You have to identify a point in time when you start counting "C"'s age. If you make it simultaneous with "B" leaving "A", then you are not actually identifying a starting point at all, because you cannot synchronize "C" with the event of "B" leaving "A". They weren't there. I pointed this out previously that you can't synchronize with remote events.

But in all explanations, we compare "C" [the home coming twin] with "A" [the staying twin] even though "C" "used to be "B"".
"A" is there at the start, and at the finish, yes. But why can't "A" be the spaceship? What if we replace the Earth with a spaceship? Where is "there", the place where "A" is all the time, if there is no absolute space?

 

[JesseM]

But in all explanations, we compare "C" [the home coming twin] with "A" [the staying twin] even though "C" "used to be "B"".
"A" is there at the start, and at the finish, yes. But why can't "A" be the spaceship? What if we replace the Earth with a spaceship? Where is "there", the place where "A" is all the time, if there is no absolute space?

You are certainly free to modify the physical situation so that A was the one who changed velocities while B did not. But then all frames will agree that A was the one who changed velocities and B did not. The point is that in the same physical situation you can't have one frame that thinks A changed velocities while B moved inertially, and a different frame which thinks B changed velocities and A moved inertially; however you choose to set up the physical situation, all inertial frames agree about which one moved inertially and which changed velocities, and they all will predict that the one who changed velocities will be younger when he reunites with the inertial twin. Velocity is relative but acceleration (change in velocity) is absolute.

 

[sylas]

This is [or should be] identical to the above situation. Why is it not? Why must I say that it is the spaceship that shifts frames?

It is not identical because you are changing who shifts frames. That MEANS it is no longer identical, of course!

Once you decide what situation to consider, involving moves and accelerations and velocities (but not gravity) SR will let you calculate all the results of all observations of any observers involved.

 

[otg]

All frames will agree that A was the one who changed velocities and B did not.

Ah this was what I needed to hear :) I think all I really needed was for someone to tell me [again].
Been quite busy lately and my mind is not really following so this struck me as really annoying since I had long accepted the twin paradox and all of relativity, especially the mathematical part. It is all too easy to fall back to pure mathematics and "leave reality behind" [to forget that the \Gamma^{\mu}_{\nu\sigma\tau} actually stands for something and just "mechanically" move indices back and forth]. But arguing that "if you calculate, you'll see" is not useful when trying to explain for non-mathematicians.

Thank you so very much for all your time and effort